If $A$ and $B$ are two invertible matrices of the same order,then $adj \,(AB)$ is equal to :-

Let $A = \begin{bmatrix} 2 & -5 \\ 3 & 1 \end{bmatrix}$. What is $f(A)$ if $f(x) = x^3 - 2x^2 - 5$?

Let $M$ be any $3 \times 3$ matrix with entries from the set $\{0, 1, 2\}$. The maximum number of such matrices,for which the sum of diagonal elements of $M^{T}M$ is $7$,is .............

Let $A = (a_{ij})$ be an $n \times n$ matrix defined by $a_{ij} = \begin{cases} k^i, & \forall i=j \\ 0, & \text{otherwise} \end{cases}$. If $m = \text{trace of } A$ and $\lim_{k \rightarrow 1} \frac{n-m}{1-k} = 171$, then the value of $n$ is:

If $\omega (\neq 1)$ is a cube root of unity,then the value of the determinant $\left| \begin{array}{ccc} 1 & 1 + i + \omega^2 & \omega^2 \\ 1 - i & -1 & \omega^2 - 1 \\ -i & -i + \omega - 1 & -1 \end{array} \right|$ is equal to

Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\det(ABA^T) = 8$ and $\det(AB^{-1}) = 8$,then $\det(BA^{-1}B^T)$ is equal to